 Original Article
 Open Access
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Finite Element Method on Shape Memory Alloy Structure and Its Applications
Chinese Journal of Mechanical Engineering volume 32, Article number: 84 (2019)
Abstract
It is significant to numerically investigate thermomechanical behaviors of shape memory alloy (SMA) structures undergoing large and uneven deformation for they are used in many engineering fields to meet special requirements. To solve the problems of convergence in the numerical simulation on thermomechanical behaviors of SMA structures by universal finite element software. This work suppose a finite element method to simulate the superelasticity and shape memory effect in the SMA structure undergoing large and uneven deformation. Two scalars, named by phasetransition modulus and equivalent stiffness, are defined to make it easy to establish and implement the finite element method for a SMA structure. An incremental constitutive equation is developed to formulate the relationship of stress, strain and temperature in a SMA material based on phasetransition modulus and equivalent stiffness. A phasetransition modulus equation is derived to describe the relationship of phasetransition modulus, stress and temperature in a SMA material during the processes of martensitic phase transition and martensitic inverse phase transition. A finite element equation is established to express the incremental relationship of nodal displacement, external force and temperature change in a finite element discrete structure of SMA. The incremental constitutive equation, phasetransition modulus equation and finite element equation compose the supposed finite element method which simulate the thermomechanical behaviors of a SMA structure. Two SMA structures, which undergo large and uneven deformation, are numerically simulated by the supposed finite element method. Results of numerical simulation show that the supposed finite element method can effectively simulate the superelasticity and shape memory effect of a SMA structure undergoing large and uneven deformation, and is suitable to act as an effective computational tool for the wide applications based on the SMA materials.
Introduction
Shape memory alloys (SMAs) have been widely used in many various engineering fields [1,2,3,4] because they possess two special thermomechanical characters, shape memory effect and superelasticity [5,6,7,8]. There are four characteristic temperatures in a SMA at a freestress state. They are named as martensitic starting temperature, indicated by M_{s}, martensitic finishing temperature, indicated by M_{f}, austenitic starting temperature, indicated by A_{s} and austenitic finishing temperature, indicated by A_{f}, respectively [9,10,11,12]. They are generally satisfied with the relationship M_{f} < M_{s} < A_{s} < A_{f}.
The shape memory effect can be expressed by the stress–strain curve in Figure 1(a), where a large nonlinear strain upon loading becomes a residual strain after unloading at a constant temperature below A_{s}. However the residual strain can be fully recovered by heating to a high temperature above A_{f}. The superelasticity can be expressed by the stress–strain curve in Figure 1(b), where the large nonlinear strain upon loading will gradually vanish during the process of unloading at a constant temperature above A_{f}. Both shape memory effect and superelasticity are the macroscopic phenomena of martensitic phase transition and martensitic inverse phase transition, which are induced by applied stress and temperature change in a SMA material [13, 14].
In order to design and analyze a SMA structure efficiently, a numerical simulation method which can effectively predict the thermomechanical behaviors of SMA should be established. As a classical numerical computational method, finite element method has been successfully used to solve many problems related to solid and/or fluid mechanics. Many universal finite element software have been successfully applied in different practical fields related to the conventional materials. Some universal finite element software, such as ABAQUS, ANSYS, and so on, have good secondary development functions, which make them can also simulate the problems related to a new materials, such as SMA through calling a user material subroutine. Some researchers [15,16,17] implemented finite element analysis on SMA structures based on the secondary development functions of ANSYS. Many researchers [18,19,20,21] implemented finite element simulation on SMA structures using the secondary development functions of ABAQUS. Although some universal finite element software can simulate the thermomechanical behaviors of SMA based on their secondary development functions, the problems of convergence often occur when a SMA structure undergoes large and uneven deformations. However many products or structures made from SMA need undergo large and uneven deformations to meet special requirements. Therefor it is necessary to develop a finite element method and its computer program to simulate the thermomechanical behaviors of a SMA structure undergoing large and uneven deformation. Zhou [22] established a finite element program to predict the martensite and plastic zone of SMA thin plate with a hole considering the plastic deformation. However, the superelasticity and shape memory effect of SMA structures during the thermodynamic loadingunloading process haven’t been numerically simulated in the work.
To avoid the problems of convergence in the numerical simulation on the thermomechanical behaviors of SMA structures with complex boundary conditions and external loads. This work suppose a finite element method to simulate the macroscopic thermomechanical responses of superelasticity and shape memory effect in a SMA structure undergoing large and uneven deformation. In order to make it easy to develop and implement the finite element method for a SMA structure, two scalars named by phasetransition modulus and equivalent stiffness are defined respectively. A concisely incremental constitutive equation formulating the relationship of stress, strain and temperature in a SMA material is derived based on phasetransition modulus and equivalent stiffness. A phasetransition modulus equation is developed to describe the evolution law of phasetransition modulus during the processes of martensitic phase transition and its inverse phase transition in a SMA material. A finite element equation is established to express the incremental relationship of nodal displacement, external force and temperature change in a finite element discrete structure of SMA. The incremental constitutive equation, phasetransition modulus equation and finite element equation compose the element finite method which simulates the thermomechanical behaviors of a SMA structure. Two SMA structures are numerically simulated by the supposed finite element method, which illustrates that it can accurately simulate the superelasticity and shape memory effect in a SMA structure undergoing large and uneven deformation. Therefor the supposed finite element method is suitable to act as an effective computational tool for the wide applications based on the SMA materials.
Incremental Constitutive Equation
Thermomechanical constitutive equation of a material is the important basis to investigate the thermomechanical behaviors of a SMA material and/or structure. Many constitutive equations have been developed to describe the thermomechanical behaviors of SMA material [23,24,25,26]. Some constitutive equations from practical viewpoints played important roles in practical applications of SMA [27,28,29,30]. In order to make it easy to establish and implement the finite element method for a SMA structure undergoing large and uneven deformation, a concisely incremental constitutive equation should be formulated to describe the thermomechanical behaviors of a SMA material.
According to the diagram of shape memory effect and superelasticity of a SMA, shown in Figure 1, the increment of strain tensor of SMA dε_{ij} can be decomposed as
where \(\text{d}\varepsilon_{ij}^{E}\), \(\text{d}\varepsilon_{ij}^{T}\) and \(\text{d}\varepsilon_{ij}^{P}\) stand for the increments of elastic strain tensor, thermal expansion strain tensor and phasetransition strain tensor, respectively.
According to generalized Hooke’s law, the increment of stress tensor of SMA dσ_{ij} reads as
where D_{ijkl} is the material stiffness tensor. The increment of thermal expansion strain tensor can be expressed as
where Λ_{ij} and dT are the thermal expansion tensor of a SMA and the increment of temperature respectively.
In this paper a SMA is assumed to be an isotropic material, therefor its stiffness tensor and thermal expansion tensor are formulated as
and
where E, v and α are elastic modulus, Poisson’s ratio and thermal expansion coefficient of SMA respectively. The elastic modulus can be expressed as
where ξ, E_{A} and E_{M} are martensitic volume fraction, austenitic elastic modulus and martensitic elastic modulus respectively.
Using Eqs. (1), (2) and (3), we can have
In this paper, the increment of phasetransition strain tensor is formulated as
where \(\bar{\sigma }\) and \(\text{d}\bar{\varepsilon }_{{}}^{P}\) are the equivalent stress and the increment of equivalent phasetransition strain respectively. According to solid mechanics, the equivalent stress reads as
The incremental relationship of equivalent stress and equivalent phasetransition strain can be expressed as
where \(\bar{H}\) is called as phasetransition modulus in this paper. The phasetransition modulus is a scalar which describes the incremental relationship of equivalent stress and equivalent phasetransition strain in a SMA material. The evolution law of phasetransition modulus is derived in Section 3.
On the other hand, the increment of equivalent stress can also be expressed as
according to Eq. (7). Using Eqs. (6), (8) and (9), we can have
From Eqs. (10) and (5), we obtain the incremental constitutive equation of a SMA, expressed as
where
In Eq. (12),
is called as equivalent stiffness in this paper. The equivalent stiffness is a scalar which depends on the stiffness tensor and stress state of a material.
The incremental constitutive equation, Eq. (11), is more conveniently used to establish and implement the finite element method for a SMA structure than those constitutive equations of SMA mentioned above, which is due to the definition of phasetransition modulus and equivalent stiffness. During the process of numerical simulation, the values of phasetransition modulus and equivalent stiffness can be calculated according to the current stress state by Eq. (12), and then the increment of stress can be calculated according to the increments of strain and temperature by Eq. (11).
PhaseTransition Modulus Equation
Based on the cosinetype martensitic phasetransition equation [9], the relationship between martensitic volume fraction and equivalent stress can be expressed as
and
during the processes of martensitic phase transition and martensitic inverse phase transition, respectively.
In Eq. (14a),
where σ_{ms} and σ_{mf} are called as martensitic starting stress and martensitic finishing stress respectively. They can be expressed by the martensitic starting temperature and martensitic finishing temperature as
where C_{M} is a material constant describing the relationship of stress and temperature during the process of martensitic phase transition in a SMA material.
In Eq. (14b),
where σ_{as} and σ_{af} are called as austenitic starting stress and austenitic finishing stress respectively. They can be expressed by the austenitic starting temperature and austenitic finishing temperature as
where C_{A} is a material constant describing the relationship of stress and temperature during the process of martensitic inverse phase transition in a SMA material.
The incremental relationship of the equivalent phasetransition strain and the martensitic volume fraction can be expressed as
According to Eqs. (8) and (16), we can have
Substituting Eq. (14a) into Eq. (17), we can express the phasetransition modulus as
during the process of martensitic phase transition in a SMA material. Substituting Eq. (14b) into Eq. (17), we can express the phasetransition modulus as
during the process of martensitic inverse phase transition in a SMA material.
Eq. (18) is the phasetransition modulus equation expressing the evolution law of phasetransition modulus, i.e., the relationship of phasetransition modulus, stress and temperature, during the processes of martensitic phasetransition and martensitic inverse phasetransition occurring in a SMA material.
Finite Element Equation
According to the stress–strain curves of SMA, shown in Figure 1, both superelasticity and shape memory effect of a SMA belong to the problem of large deformation. Therefor the TotalLagrange method suitable for the problem of large deformation is used to develop the finite element equation for a SMA structure. Considering one point in a loaded body, the displacement, GreenLagrange strain and PiolaKirchhoff stress at the time t and t + Δt are respectively expressed as
and \(u_{i} + \Delta u_{i} , \, \varepsilon_{ij} + \Delta \varepsilon_{ij} , \, \sigma_{ij} + \Delta \sigma_{ij}\).
They can also be respectively expressed as
and
by the form of matrix. The increment of strain can be expressed as
which is also expressed as
by the form of matrix, where Δε_{L0} is corresponding with the first and second terms, Δε_{L1} is corresponding with the third and fourth terms, and Δε_{N} is corresponding with the fifth term in the right hand of equality sign in Eq. (20) respectively.
At the time t + Δt, the virtual work equation of a finite element discrete structure reads as
where δ is the symbol of variation, V_{e} and S_{e} express the volume and surface of an element, and f and q stand for body force array and surface force array respectively. The increment of displacement array at one point in an element can be expressed by the increment of nodal displacement array of the element as
where \(\Delta \varvec{u}^{e}\) is the increment of nodal displacement array of the element and N is the shape function matrix of the element, respectively.
Operating the variation operation on Eq. (22) leads to
The increment of strain array is formulated by the increment of nodal displacement array as
where B is the strain matrix of element. It includes B_{L0}, B_{L1} and B_{N}, which are with respect to the Δε_{L0}, Δε_{L1} and Δε_{N} in Eq. (20), respectively. According to Eqs. (19) and (24), if the higher order small term is ignored, the variation of the increment of strain can be expressed as
In order to establish finite element equation for a SMA structure, we express the incremental constitutive equation, Eq. (11), with the form of matrix as
Substituting Eqs. (22)–(26) into Eq. (21) and carrying out the necessary derivations and reductions, we can have
where
The ΔU and δΔU in Eq. (28) denote the increment of nodal displacement array and its virtual displacement array of the finite element discrete structure respectively. The P_{F} in Eq. (28e) is the equivalent nodal loads array reduced from external force. The P_{T} in Eq. (28d) is the equivalent nodal loads array reduced from the change of temperature. The P_{σ} in Eq. (28a) is the equivalent nodal loads array reduced from nonequilibrium force.
The nodal virtual displacement array δΔU should be arbitrary, so Eq. (27) can be arranged and reduced as
where K_{L} and K_{N} express structural stiffness matrices which relate with small deformation and large deformation respectively. The nonlinear equation, Eq. (29), is the finite element equation which describes the relationship of the nodal displacement, external force and temperature change in a finite element discrete structure of SMA. This nonlinear equation is solved by the NewtonRaphson method during the processes of finite element simulations on the SMA structures in this paper.
The incremental constitutive equation, Eq. (11), the phasetransition modulus equation, Eq. (18), and the finite element equation, Eq. (29), compose the supposed finite element method, which simulate the thermomechanical macroscopic response of superelasticity and shape memory effect occurring in a SMA structure.
Applications
In order to operate finite element simulations on the thermomechanical response of a SMA structure using the supposed finite element method mentioned above, a MATLAB program is compiled based on finite element equation, Eq. (29), incremental constitutive equation, Eq. (11), and phasetransition modulus equation, Eq. (18). A uniform tensile SMA bar, shown in Figure 2(a), is numerically simulated to verify the MATLAB program. The material parameters of SMA [23] used for the numerical simulations are listed in Table 1. Figure 2(b) plots the finite element mesh of the SMA bar, which include 42 plane strain fournode elements. The numerical results at integral point 1 of the shaded element, shown in Figure 2(b), are used to compare with the analytic solutions [9] to illustrate the validity of the supposed finite element method in this paper.
Figure 3(a) plots the stress–strain curve of the integral point 1, shown in Figure 2(b), at a constant temperature above austenitic finishing temperature A_{f}. The large nonlinear phasetransition strain produced in the loading process is fully recovered during the unloading process, which illustrates the superelasticity of a SMA material. Figure 3(b) shows the stress–strain curve of the integral point 1, shown in Figure 2(b), at a constant temperature below austenitic finishing temperature A_{f}. The large nonlinear phasetransition strain produced in the loading process becomes residual strain after the unloading process. The residual strain can be fully recovered upon heating to a high temperature above A_{f}, which illustrates the shape memory effect of a SMA material.
According to the curves in Figure 3(a) and Figure 3(b), the results from the supposed finite element method in this paper have a good agreement with that from the analytic solutions [9]. Therefore the supposed finite element method in this paper can effectively simulate the thermomechanical macroscopic responses of superelasticity and shape memory effect occurring in a SMA structure.
Thermomechanical Behaviors of SMA Bar
Figure 4 shows a deformation mechanism for a SMA bar, which includes two punch heads, one die and one support. The moments of force applied on the punch head make the punch head rotate around the rotation point and generate a large and uneven bending deformation in the SMA bar. The SMA bar has length of 28 mm, width of 2 mm and thickness of 4 mm. The radius of arc in the punch head is 6 mm, and the radius of the head in the die is 8 mm. During the numerical simulations the SMA bar is meshed by the 4node plane strain element, and the total number of element is 224. The punch head and die are assumed to be rigid body. The frictionfree contacts are set between punch head and SMA bar and between punch head and SMA bar. The material constants of SMA bar are listed in Table 1.
Figure 5 plots the relational curve between the moment of force and the rotation angle of punch head at a constant temperature of 317 K. During the loading process there is a nonlinear curve segment occurring, which is the results of martensitic phase transition occurring in some materials of the SMA bar. During the unloading process there is a nonlinear curve segment occurring, which is the result of martensitic inverse phase transition occurring in the materials having experienced martensitic phase transition in the loading process. There is a hysteresis loop occurring in the curve, which illustrates the superelasticity of the SMA bar. Figure 6 plots the relational curve between the moment of force and the rotation angle of punch head at a constant temperature of 300 K. During the loading process there is also a nonlinear curve segment occurring, which is also the result of martensitic phase transition occurring in some materials of SMA bar. However there is not a hysteresis loop occurring in the curve after the unloading process.
Figure 7 shows the shape changing process of SMA bar at a constant temperature of 317 K in the loadingunloading cycle. The deformed shapes of SMA bar at the same moment of force during the processes of loading and unloading are different. This is because the loadingpath does not coincide with the unloadingpath, which is results from the hysteresis loop shown in Figure 5. Figure 8 shows the shape changing process of SMA bar in the loadingunloadingheating cycle. The deformation of SMA bar, which is induced by the martensitic phase transition upon loading, becomes the residual deformation after the unloading process. The residual deformation in the SMA bar is fully recovered by heating it from 300 K to 320 K, which illustrates the shape memory effect of SMA bar.
Thermomechanical Behaviors of SMA Cirque
Figure 9 shows a deformation mechanism for a SMA cirque, which includes a punch head and a support. The applied force makes the punch head move down and produce large and uneven deformation in the SMA cirque. The radius of punch head bottom is 2 mm. The inside radius and outer radius of SMA cirque are 10 mm and 12 mm respectively. The thickness of SMA cirque is 4 mm. During the numerical simulations the SMA cirque is meshed by the 4node plane strain element, and the total number of element is 544. The punch head is assumed to be a rigid body. The frictionfree contacts are set between the punch head and SMA cirque. The material parameters of SMA cirque are listed in Table 1.
Figure 10 plots the relational curve between the applied force and the displacement of punch head at a constant temperature of 320 K. During the loading process there is a nonlinear curve segment due to the martensitic phase transition occurring in some materials of the SMA cirque. During the unloading process there is also a nonlinear curve segment occurring, which is the result of martensitic inverse phase transition occurring in the materials having experienced martensitic phase transition in the loading process. There is a hysteresis loop occurring in the curve after unloading, which illustrates the superelasticity of SMA cirque. Figure 11 plots the relational curve between the applied force and the displacement of punch head at a constant temperature of 295 K. During the loading process there is a nonlinear curve segment due to the martensitic phase transition occurring in some materials of SMA cirque. There is not a hysteresis loop occurring in the curve after unloading.
Figure 12 shows the shape changing process of SMA cirque in a loadingunloading cycle at a constant temperature of 320 K. The deformed shapes of SMA cirque corresponding with the same applied force during the processes of loading and unloading are different because the loadingpath does not coincide with the unloadingpath, as shown in Figure 10. Figure 13 shows the shape changing process of SMA cirque in the loadingunloadingheating cycle. The deformation of SMA cirque, which is induced by the martensitic phase transition upon loading, becomes the residual deformation after unloading. However, the residual deformation in the SMA cirque is fully recovered by heating it from 295 K to 320 K, which illustrates the shape memory effect of SMA cirque.
On the whole, the supposed finite element method has good accuracy and convergence in the simulation on both superelasticity and shape memory effect in a SMA structure, and is suitable to be an effective computational tool for the wide applications based on a SMA material.
Conclusions
Two scalars, phasetransition modulus and equivalent stiffness, are defined to establish and implement the finite element method. The superelasticity and shape memory effect of SMA bar and cirque are respectively simulated by the supposed finite element method. Accordingly, several conclusions are presented as follows.

1)
The concisely incremental constitutive equation describing the relationship of stress, strain and temperature in a SMA material is developed based on phasetransition modulus and equivalent stiffness.

2)
The phasetransition modulus equation expressing the relationship of phasetransition modulus, stress and temperature during the processes of martensitic phase transition and martensitic inverse phase transition in a SMA material is presented.

3)
The finite element equation formulating the incremental relationship of nodal displacement, external force and temperature change in a finite element discrete structure is established to simulate the thermomechanical behaviors of a SMA structure.

4)
The supposed finite element method, which includes the incremental constitutive equation, phasetransition modulus equation and finite element equation, can effectively simulate both processes of superelasticity and shape memory effect in a SMA structure, and is suitable to act as an effective computational tool for the wide applications based on a SMA material.
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Authors’ Contributions
BZ was in charge of the whole trial; ZK and ZW compiled the calculation programs and wrote Section 5; SX wrote Section 4. All authors read and approved the final manuscript.
Authors’ Information
Bo Zhou, born in 1972, is currently a professor and a doctoral supervisor at Department of Engineering Mechanics, College of Pipeline and Civil Engineering, China University of Petroleum, China. His research interests include intelligent materials and structural mechanics, computational mechanics and engineering mechanics of oil and gas wells.
Zetian Kang, born in 1992, is currently a PhD candidate at China University of Petroleum, China. He received his bachelor degree from China University of Petroleum, in 2014. His research interest is intelligent materials and structural mechanics.
Zhiyong Wang, born in 1992, is currently a graduate student at China University of Petroleum, China. He received his bachelor degree from China University of Petroleum, China, in 2017. His research interest is intelligent materials and structural mechanics.
Shifeng Xue, born in 1963, is currently a professor and a doctoral supervisor at Department of Engineering Mechanics, College of Pipeline and Civil Engineering, China University of Petroleum, China. His research interests include computational mechanics, fracture mechanics and engineering mechanics of oil and gas wells.
Acknowledgements
The authors of this paper sincerely thank to the supports from the China University of Petroleum (East China).
Competing Interests
The authors declare that they have no competing interests.
Funding
Supported by National Key Research and Development Program of China (Grant No. 2017YFC0307604), and the Talent Foundation of China University of Petroleum (Grant No. Y1215042).
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Zhou, B., Kang, Z., Wang, Z. et al. Finite Element Method on Shape Memory Alloy Structure and Its Applications. Chin. J. Mech. Eng. 32, 84 (2019). https://doi.org/10.1186/s1003301904013
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Keywords
 Shape memory alloy
 Incremental constitutive equation
 Finite element equation
 Phasetransition modulus
 Applications